Answer
We use the definition of differentiable to define the local linearity for functions of two variables.
Work Step by Step
Using the definition of differentiable, we define the local linearity for functions of two variables:
Definition. A function of two variables $f\left( {x,y} \right)$ is locally linear if
$f\left( {x,y} \right) = L\left( {x,y} \right) + e\left( {x,y} \right)$
where $e\left( {x,y} \right)$ satisfies
$\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {a,b} \right)} \frac{{e\left( {x,y} \right)}}{{\sqrt {{{\left( {x - a} \right)}^2} + {{\left( {y - b} \right)}^2}} }} = 0$
and
$L\left( {x,y} \right) = f\left( {a,b} \right) + {f_x}\left( {a,b} \right)\left( {x - a} \right) + {f_y}\left( {a,b} \right)\left( {y - b} \right)$
